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Hägg, Linus

Open this publication in new window or tab >>On minimum length scale control in density based topology optimization### Hägg, Linus

### Wadbro, Eddie

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##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer, 2018
##### Keywords

topology optimization, nonlinear filters, size control, mathematical morphology
##### National Category

Computer Sciences Computational Mathematics
##### Identifiers

urn:nbn:se:umu:diva-151378 (URN)10.1007/s00158-018-1944-0 (DOI)000441847800010 ()
#####

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##### Funder

Swedish Foundation for Strategic Research , AM13-0029Swedish Research Council, 621-3706
Available from: 2018-09-06 Created: 2018-09-06 Last updated: 2018-09-06Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Umeå University, Faculty of Science and Technology, Department of Computing Science.

The archetypical topology optimization problem concerns designing the layout of material within a given region of space so that some performance measure is extremized. To improve manufacturability and reduce manufacturing costs, restrictions on the possible layouts may be imposed. Among such restrictions, constraining the minimum length scales of different regions of the design has a significant place. Within the density filter based topology optimization framework the most commonly used definition is that a region has a minimum length scale not less than D if any point within that region lies within a sphere with diameter D > 0 that is completely contained in the region. In this paper, we propose a variant of this minimum length scale definition for subsets of a convex (possibly bounded) domain We show that sets with positive minimum length scale are characterized as being morphologically open. As a corollary, we find that sets where both the interior and the exterior have positive minimum length scales are characterized as being simultaneously morphologically open and (essentially) morphologically closed. For binary designs in the discretized setting, the latter translates to that the opening of the design should equal the closing of the design. To demonstrate the capability of the developed theory, we devise a method that heuristically promotes designs that are binary and have positive minimum length scales (possibly measured in different norms) on both phases for minimum compliance problems. The obtained designs are almost binary and possess minimum length scales on both phases.

Open this publication in new window or tab >>Topology optimization of compact wideband coaxial-to-waveguide transitions with minimum-size control### Hassan, Emadeldeen

### Wadbro, Eddie

### Hägg, Linus

### Berggren, Martin

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_otherAuthors",{id:"formSmash:j_idt184:1:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_otherAuthors",multiple:true}); 2018 (English)In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 57, no 4, p. 1765-1777Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

New York: Springer, 2018
##### Keywords

Maxwell's equations, sensitivity analysis, optimization, waveguide
##### National Category

Communication Systems Computer Sciences
##### Identifiers

urn:nbn:se:umu:diva-146663 (URN)10.1007/s00158-017-1844-8 (DOI)000430101600022 ()
#####

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Available from: 2018-04-16 Created: 2018-04-16 Last updated: 2018-06-09Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Computing Science. Department of Electronics and Electrical Communications, Menoufia University, Menouf, Egypt.

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.

This paper presents a density-based topology optimization approach to design compact wideband coaxial-to-waveguide transitions. The underlying optimization problem shows a strong self penalization towards binary solutions, which entails mesh-dependent designs that generally exhibit poor performance. To address the self penalization issue, we develop a filtering approach that consists of two phases. The first phase aims to relax the self penalization by using a sequence of linear filters. The second phase relies on nonlinear filters and aims to obtain binary solutions and to impose minimum-size control on the final design. We present results for optimizing compact transitions between a 50-Ohm coaxial cable and a standard WR90 waveguide operating in the X-band (8-12 GHz).

Open this publication in new window or tab >>Nonlinear filters in topology optimization: existence of solutions and efficient implementation for minimum compliance problems### Hägg, Linus

### Wadbro, Eddie

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_some",{id:"formSmash:j_idt184:2:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_otherAuthors",{id:"formSmash:j_idt184:2:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_otherAuthors",multiple:true}); 2017 (English)In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 55, no 3, p. 1017-1028Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Topology optimization, Regularization, Nonlinear filters, Existence of solutions, Large-scale problems
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:umu:diva-133354 (URN)10.1007/s00158-016-1553-8 (DOI)000398114200019 ()
#####

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##### Funder

Swedish Foundation for Strategic Research , AM13-0029Swedish Research Council, 621-3706
Available from: 2017-04-06 Created: 2017-04-06 Last updated: 2018-06-09Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Material distribution topology optimization problems are generally ill-posed if no restriction or regularization method is used. To deal with these issues, filtering procedures are routinely applied. In a recent paper, we presented a framework that encompasses the vast majority of currently available density filters. In this paper, we show that these nonlinear filters ensure existence of solutions to a continuous version of the minimum 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 large-scale numerical experiments illustrating some characteristics of these cascaded nonlinear filters.

Open this publication in new window or tab >>Nonlinear filters in topology optimization: existence of solutions and efficient implementation for minimal compliance problems### Hägg, Linus

### Wadbro, Eddie

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_some",{id:"formSmash:j_idt184:3:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_otherAuthors",{id:"formSmash:j_idt184:3:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_otherAuthors",multiple:true}); 2016 (English)Report (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Umeå: Umeå universitet, 2016. p. 16
##### Series

Report / UMINF, ISSN 0348-0542 ; 16.01
##### National Category

Computer Sciences
##### Identifiers

urn:nbn:se:umu:diva-115934 (URN)
#####

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##### Funder

Swedish Research Council, 621-3706Swedish Foundation for Strategic Research , AM13-0029
Available from: 2016-02-08 Created: 2016-02-08 Last updated: 2018-06-07Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Umeå University, Faculty of Science and Technology, Department of Computing Science.

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.

Open this publication in new window or tab >>Quasi-Arithmetic Filters for Topology Optimization### Hägg, Linus

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##### Alternative title[sv]

Kvasiaritmetiska filter för topologioptimering
##### Abstract [en]

##### Place, publisher, year, edition, pages

Umeå: Umeå Universitet, 2016. p. 23
##### Series

Report / UMINF, ISSN 0348-0542 ; 16.04
##### National Category

Computational Mathematics Computer Sciences
##### Identifiers

urn:nbn:se:umu:diva-116983 (URN)978-91-7601-409-7 (ISBN)
##### Presentation

2016-02-19, Umeå universitet, Umeå, 09:00
#####

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##### Supervisors

### Berggren, Martin

### Wadbro, Eddie

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##### Funder

eSSENCE - An eScience CollaborationSwedish Research Council, 621-3706
Available from: 2016-02-25 Created: 2016-02-16 Last updated: 2018-06-07Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Computing Science.

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.

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Open this publication in new window or tab >>1D-model of the interaction between a stack of wood and an imposed electromagnetic wave### Hägg, Linus

### Wadbro, Eddie

### Noreland, Daniel

### Berggren, Martin

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##### Abstract [en]

##### Keywords

moisture content, wood, electromagnetic, non-invasive measurement
##### National Category

Computational Mathematics Other Environmental Engineering
##### Identifiers

urn:nbn:se:umu:diva-112582 (URN)
#####

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##### Funder

eSSENCE - An eScience Collaboration
Available from: 2015-12-11 Created: 2015-12-10 Last updated: 2018-06-07Bibliographically approved

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.

Umeå University, Faculty of Science and Technology, Department of Computing Science.

We have developed and investigated a 1D-model for the interaction between a stack of wood and an impinging electromagnetic field. Maxwell's equations are used to model the electromagnetic interaction and each layer in a stack of boards has been modeled as a homogenous lossy dielectric slab. The main reason for developing this model has been to investigate the possibility of measuring the moisture content of wood inside a drying kiln using electromagnetic waves. Our investigations show that it is in principle possible to measure the moisture content, since the electromagnetic field is sensitive to changes in the moisture content of the wood. We also show that it might be possible to measure the average moisture content, without detailed knowledge of the distribution of moisture content between different boards.

Open this publication in new window or tab >>On quasi-arithmetic mean based filters and their fast evaluation for large-scale topology optimization### Wadbro, Eddie

### Hägg, Linus

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_some",{id:"formSmash:j_idt184:6:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_otherAuthors",{id:"formSmash:j_idt184:6:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_otherAuthors",multiple:true}); 2015 (English)In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 52, no 5, p. 879-888Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer, 2015
##### Keywords

Topology optimization, Regularization, Filters, Fast algorithm, Large-scale problems
##### National Category

Computer Sciences
##### Identifiers

urn:nbn:se:umu:diva-114034 (URN)10.1007/s00158-015-1273-5 (DOI)000366590800003 ()
#####

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Available from: 2016-01-11 Created: 2016-01-11 Last updated: 2018-06-07Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Umeå University, Faculty of Science and Technology, Department of Computing Science.

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.